STA 237 Probability Statistics and Data Analysis Assignment

STA 237 Probability Statistics and Data Analysis Assignment

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STA 237 Probability Statistics and Data Analysis Assignment

Question:2 

Suppose you are working for TD’s car insurance program. You are helping the leadership team to set the premium rate for the upcoming year. You are told that there are currently 1 million customers who have their car insurance through TD.

Each customer has a 0.05% of getting into a major accident and a 1% chance of getting into a minor accident. To keep our calculations simple, let’s assume a client can only be involved in a maximum of one accident in a year.

Let X be the amount in dollars that a customer is required to pay to repair his/her vehicle anytime it is involved in a minor accident. And let Y be the amount in dollars that a customer is required to pay anytime it is involved in a major accident.

There is a thousand dollar deductible in every insurance plan, which means TD will not pay anything to the customer if the repair cost is less or equal to $1000. And for any repair cost that is above $1000, the customer pays the first $1000 and TD will pay the rest.

Let X’s probability density function be

f(x) = 1

3000e≠ 1 3000x; x Ø 0

and Y ’s probability density function be

f(y) = 1

20000e≠ 1

20000 y; y Ø 0

Let Z be the dollar value that TD needs to pay to each customer during the year.

a)[2 points] Write a function in R that will mimic this whole scenario for one of the customer (i.e. it will simulate if the customer will get into an accident or not, based on the type of accident it will simulate the repair cost and depending on the cost it will calculate the amount TD will have to pay).  STA 237 Probability Statistics and Data Analysis Assignment

b)[0.5 point] Use your function to calculate an estimate of E[Z].

c)[0.5 point] If TD wants to cover the expected TOTAL cost(the total from the claims of all the customers) and a fixed operating cost of 5 million dollars, how much should the monthly premium be for each of their customer?

Expected output: 

(a) Code of an R function. (b) Few lines of R code with one line of R output. (c) Numeric calculations and/or R code.

© 2020 Shahriar Shams, University of Toronto Page 3 © 2020 Shahriar Shams, University of Toronto Page 3 © 2020 Shahriar Shams, University of Toronto Page 3

Question:3 

In Lecture-6 (Book chapter 6.2), we learned how to generate random numbers from certain distributions using random numbers from Unif(0,1). You will do two examples here.

(a)[1 point] Suppose X is a continuous random variable with pdf

f(x)=4x3 ; 0 Æ x Æ 1

• Use 100000 random numbers from Unif(0,1) and the distribution function of X to convert each of these uniform random numbers into X.
• Plot a density histogram of the X values that you generated to check if it really looks like the given pdf.

(b)[2 points] Using the same idea as part(a), generate 100000 random numbers from this following pdf

f(x) =

Y_]_[

1x8 0 8 ;0 Æ x < 2 ;2 Æ x Æ 4 ;otherwise 

1x8 0 8 ;0 Æ x < 2 ;2 Æ x Æ 4 ;otherwise 

Start by deriving the distribution function. Make sure to check the density histogram once you have generated X.

Expected output: 

(a) Distribution function of X, an R code and one plot. (b) Distribution function of X, an R code and one plot.  STA 237 Probability Statistics and Data Analysis Assignment

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